Foliations and Diffeomorphism Groups — Question 31

Definitions and mathematical statement

The supplied question is: Do volume preserving actions of surface groups on ${\mathbb{R}}^2$ satisfy the Milnor--Wood inequality? The record does not specify the class of surfaces, the regularity of the actions, the preserved volume, or the precise inequality.

For a conditional formulation, suppose that the intended surface is a closed connected oriented surface $\Sigma_g$ of genus $g\geq 1$, and that the intended actions are smooth actions preserving a specified smooth positive area form $\omega$ on ${\mathbb{R}}^2$. Write \[ \operatorname{Diff}({\mathbb{R}}^2,\omega)=\{f\in\operatorname{Diff}^{\infty}({\mathbb{R}}^2):f^*\omega=\omega\}. \] An action is then a homomorphism $\rho:\pi_1(\Sigma_g)\to\operatorname{Diff}({\mathbb{R}}^2,\omega)$. Its associated flat oriented plane bundle is \[ E_\rho=(\widetilde{\Sigma}_g\times{\mathbb{R}}^2)/\pi_1(\Sigma_g),\qquad \gamma\cdot(x,v)=(\gamma x,\rho(\gamma)(v)). \] Let $e(E_\rho)\in H^2(\Sigma_g;{\mathbb{Z}})$ denote its Euler class as an oriented topological plane bundle, and define its Euler number by \[ \operatorname{eu}(\rho)=\langle e(E_\rho),[\Sigma_g]\rangle\in{\mathbb{Z}}. \] Under these explicitly conditional assumptions, the intended question would ask whether every such $\rho$ satisfies \[ |\operatorname{eu}(\rho)|\leq B(g), \] for the intended Milnor--Wood bound $B(g)$. The supplied record does not identify this bound. In particular, the circle-action bound $B(g)=2g-2$ and the bound $B(g)=g-1$ for flat oriented real rank-two vector bundles are distinct conventions relevant to this terminology; the record does not determine which, if either, is intended here. The surface, smoothness, and area-form assumptions above are also conditional and are not supplied hypotheses. Thus this template does not constitute a uniquely determined formulation of the source question.

Short English statement

The question asks whether preserving volume forces surface-group actions on ${\mathbb{R}}^2$ to obey an Euler-number bound of Milnor--Wood type. The supplied record leaves the action category, surface hypotheses, preserved volume, and numerical bound unspecified, so a definitive quantified statement requires clarification from the source.

Sources

• Local UnsolvedMath registry, permanent ID $800090$, exact source problem code catalogue entry, title Foliations and Diffeomorphism Groups --- Question 31. The registry attributes the problem to Nariman and Yazdi, with proposed year $2025$, and labels it open; this is a supplied status label, not an independently verified assessment.
• Nariman and Yazdi, Problem List on Foliations and Diffeomorphism Groups ($2025$), Question 31. Supplied source URL: https://arxiv.org/abs/2503.18274. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance details; the source has not been independently accessed or verified for this draft.